Repositories are Internet servers which hold software packages. These can be accessed by your computer. When you install Mint, it already comes configured with a list of sources. The list can be found at: /etc/apt/sources.lst

When you fire up Synaptic, it accesses these servers. Hit reload and you get the latest software available. It will tell you what's installed, what's not, and what you can upgrade. You can then choose the software you want to add, remove, or upgrade. Since you seem new to this, as I am, it's probably best to leave the list of repositories as it is, unless advised to add a new one by Clem or someone who works on the project. Meantime, there's a wealth of information available through Google, or pages such as Ubuntu Wiki and forums.

Since Mint is based on Ubuntu 6.10, we have access to their repos. Whether Mint will benefit from the Linspire/Ubuntu deal depends on whether Clem decides to keep Mint based on Ubuntu. Whether this will be a benefit is still being argued by the cognoscenti. I am not one, so no opinion here.